Theorem funmo 5904

Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Assertion

Ref	Expression
funmo	|- ( Fun F -> E* y A F y )

Distinct variable groups:   y, A   y, F

Proof of Theorem funmo

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun6 5903	. . . . . 6 |- ( Fun F <-> ( Rel F /\ A. x E* y x F y ) )
2	1	simplbi 476	. . . . 5 |- ( Fun F -> Rel F )
3		brrelex 5156	. . . . . 6 |- ( ( Rel F /\ A F y ) -> A e. _V )
4	3	ex 450	. . . . 5 |- ( Rel F -> ( A F y -> A e. _V ) )
5	2, 4	syl 17	. . . 4 |- ( Fun F -> ( A F y -> A e. _V ) )
6	5	ancrd 577	. . 3 |- ( Fun F -> ( A F y -> ( A e. _V /\ A F y ) ) )
7	6	alrimiv 1855	. 2 |- ( Fun F -> A. y ( A F y -> ( A e. _V /\ A F y ) ) )
8		breq1 4656	. . . . . . 7 |- ( x = A -> ( x F y <-> A F y ) )
9	8	mobidv 2491	. . . . . 6 |- ( x = A -> ( E* y x F y <-> E* y A F y ) )
10	9	imbi2d 330	. . . . 5 |- ( x = A -> ( ( Fun F -> E* y x F y ) <-> ( Fun F -> E* y A F y ) ) )
11	1	simprbi 480	. . . . . 6 |- ( Fun F -> A. x E* y x F y )
12	11	19.21bi 2059	. . . . 5 |- ( Fun F -> E* y x F y )
13	10, 12	vtoclg 3266	. . . 4 |- ( A e. _V -> ( Fun F -> E* y A F y ) )
14	13	com12 32	. . 3 |- ( Fun F -> ( A e. _V -> E* y A F y ) )
15		moanimv 2531	. . 3 |- ( E* y ( A e. _V /\ A F y ) <-> ( A e. _V -> E* y A F y ) )
16	14, 15	sylibr 224	. 2 |- ( Fun F -> E* y ( A e. _V /\ A F y ) )
17		moim 2519	. 2 |- ( A. y ( A F y -> ( A e. _V /\ A F y ) ) -> ( E* y ( A e. _V /\ A F y ) -> E* y A F y ) )
18	7, 16, 17	sylc 65	1 |- ( Fun F -> E* y A F y )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   _Vcvv 3200   class class class wbr 4653   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  funeu  5913  funco  5928  fununmo  5933  imadif  5973  fneu  5995  dff3  6372  shftfn  13813